# How do you evaluate the definite integral by the limit definition given #int 6dx# from [4,10]?

##### 1 Answer

#### Explanation:

**Geometric Approach**

This is the equivalent of drawing a rectangle whose boundaries are the line

This rectangle would therefore have a length of

Since the definite integral is in fact a measure of area, this is the answer.

**Using the limit definition**

The limit definition of a definite integral is

The parameter

Therefore:

#=lim_(n->oo) sum_(i = 1)^n f (4+(6i)/n) xx 6/n#

Since

#=lim_(n->oo) sum_(i = 1)^n 6 xx 6/n#

#=lim_(n->oo) sum_(i = 1)^n 36/n#

Use the formula

#=lim_(n-> oo) n(36)/n#

#=lim_(n->oo) 36#

#= 36#

Same answer!

**Calculus Approach**

We integrate using the formula

We now rewrite in proper notation, and evaluate using the second fundamental theorem of calculus, which is that

Same answer as before, just using a different method.

Hopefully this helps!